Category Archives: research

It is a celebrated result of John Dixon, (The probability of generating the symmetric group, (subscription) Math. Z. 110 (1969), 199–205.) that if one choose two random permutations in the symmetric group , uniformly (i.e. each with probability ), and independently, the probability that the two permutations generate the whole group tends to as . It is clear that this probability will never be greater than , since there is a probability that the two permutations will both be even, in which case you could only generate, at most, the alternating group. Interestingly enough, Dixon’s paper covers this possibility, and he actually shows that the probability that two permutations generate the alternating group tends to as .

Equivalently, if two random elements of the alternating group are chosen uniformly and randomly, the probability that they generate the group tends to as . This leads me to my question — what is the probability that the -regular Cayley graph with generators is not Hamiltonian, as ?

Showing that this probability is bounded away from would provide a counterexample for a notorious problem about vertex-transitive graphs. So we might expect that this is hard. But is it even possible that it is true, or is there some obvious reason that such graphs will tend to be Hamiltonian?

Another approach in the same spirit would be computational rather than asymptotic. Suppose we look at thousands of random Cayley graphs on the alternating groups and , for example. It is straightforward to check that they are connected. Is it within reach for a cleverly designed algorithm on modern computers to conclusively rule out Hamiltonicity for a -regular graph on or vertices? I would also be happy with a computer-aided proof that the conjecture is false.

Historical note: It is called the Lovász conjecture, even though he just asked the question (and perhaps conjectured the other way). I am under the impression that some prominent people in this field have felt that the answer should be no. In particular Babai does not believe it.

Last spring I saw a great colloquium talk on packing regular tetrahedra in space by Jeffrey Lagarias. He pointed out that in some sense the problem goes back to Aristotle, who apparently claimed that they tile space. Since Aristotle was thought to be infallible, this was repeated throughout the ages until someone (maybe Minkowski?) noticed that they actually don’t.

John Conway and Sal Torquato considered various quantitative questions about packing, tiling, and covering, and in particular asked about the densest packing of tetrahedra in space. They optimized over a very special kind of periodic packing, and in the densest packing they found, the tetrahedra take up about 72% of space.

Compare this to the densest packing of spheres in space, which take up about 74%. If Conway and Torquato’s example was actually the densest packing of tetrahedra, it would be a counterexample to Ulam’s conjecture that the sphere is the worst case scenario for packing.

But a series of papers improving the bound followed, and as of early 2010 the record is held by Chen, Engel, and Glotzer with a packing fraction of 85.63%.

I want to advertise two attractive open problems related to this.

(1) Good upper bounds on tetrahedron packing.

At the time of the colloquium talk I saw several months ago, it seemed that despite a whole host of papers improving the lower bound on tetrahedron packing, there was no upper bound in the literature. Since then Gravel, Elser, and Kallus posted a paper on the arXiv which gives an upper bound. This is very cool, but the upper bound on density they give is something like , so there is still a lot of room for improvement.

(2) Packing tetrahedra in a sphere.

As far as I know, even the following problem is open. Let’s make our lives easier by discretizing the problem and we simply ask how many tetrahedra we can pack in a sphere. Okay, let’s make it even easier: the edge length of each of the tetrahedra is the same as the radius of the sphere. Even easier: every one of the tetrahedra has to have one corner at the center of the sphere. Now how many tetrahedra can you pack in the sphere?

It is fairly clear that you can get 20 tetrahedra in the sphere, since the edge length of the icosahedron is just slightly longer than the radius of its circumscribed sphere. By comparing the volume of the regular tetrahedron to the volume of the sphere, we get a trivial upper bound of 35 tetrahedra. But by comparing surface area instead, we get an upper bound of 22 tetrahedra.

There is apparently a folklore conjecture that 20 tetrahedra is the right answer, so proving this comes down to ruling out 21 or 22. To rule out 21 seems like a nonlinear optimization problem in some 63-dimensional space.

I’d guess that this is within the realm of computation if someone made some clever reductions. Oleg Musin settled the question of the kissing number in 4-dimensional space in 2003. To rule out kissing number of 25 is essentially optimizing some function over a 75-dimensional space. This sounds a little bit daunting, but it is apparently much easier than Thomas Hales’s proof of the Kepler conjecture. (For a nice survey of this work, see this article by Pfender and Ziegler.)

Eric Babson, Chris Hoffman, and I recently posted major revisions of our preprint, “The fundamental group of random 2-complexes” to the arXiv. This article will appear in Journal of the American Mathematical Society. This note is intended to be a high level summary of the main result, with a few words about the techniques.

The Erdős–Rényi random graph is the probability space on all graphs with vertex set , with edges included with probability , independently. Frequently and , and we say that asymptotically almost surely (a.a.s) has property if as .

A seminal result of Erdős and Rényi is that is a sharp threshold for connectivity. In particular if , then is a.a.s. connected, and if , then is a.a.s. disconnected.

Linial and Meshulam showed that is a sharp threshold for vanishing of first homology . (Here the coefficients are over . This was generalized to for all by Meshulam and Wallach.) In other words, once is much larger than , every (one-dimensional) cycle is the boundary of some two-dimensional subcomplex.

Babson, Hoffman, and I showed that the threshold for vanishing of is much larger: up to some log terms, the threshold is . In other words, you must add a lot more random two-dimensional faces before every cycle is the boundary of not any just any subcomplex, but the boundary of the continuous image of a topological disk. A precise statement is as follows.

Main result Let be arbitrary but constant. If then , and if then , asymptotically almost surely.

It is relatively straightforward to show that when is much larger than , a.a.s. . Almost all of the work in the paper is showing that when is much smaller than a.a.s. . Our methods depend heavily on geometric group theory, and on the way to showing that is non-vanishing, we must show first that it is hyperbolic in the sense of Gromov.

Proving this involves some intermediate results which do not involve randomness at all, and which may be of independent interest in topological combinatorics. In particular, we must characterize the topology of sufficiently sparse two-dimensional simplicial complexes. The precise statement is as follows:

Theorem. If is a finite simplicial complex such that for every subcomplex , then is homotopy equivalent to a wedge of circle, spheres, and projective planes.

(Here denotes the number of -dimensional faces.)

Corollary. With hypothesis as above, the fundamental group is isomorphic to a free product , for some number of ‘s and ‘s.

It is relatively easy to check that if then with high probability subcomplexes of on a bounded number of vertices satisfy the hypothesis of this theorem. (Of course itself does not, since it has and roughly as approaches .)

But the corollary gives us that the fundamental group of small subcomplexes is hyperbolic, and then Gromov’s local-to-global principle allows us to patch these together to get that is hyperbolic as well.
This gives a linear isoperimetric inequality on which we can “lift” to a linear isoperimetric inequality on .

But if is simply connected and satisfies a linear isoperimetric inequality, then that would imply that every -cycle is contractible using a bounded number of triangles, but this is easy to rule out with a first-moment argument.

There are a number of technical details that I am omitting here, but hopefully this at least gives the flavor of the argument.

An attractive open problem in this area is to identify the threshold for vanishing of . It is tempting to think that , since this is the threshold for vanishing of for every integer . This argument would work for any fixed simplicial complex but the argument doesn’t apply in the limit; Meshulam and Wallach’s result holds for fixed as , so in particular it does not rule out torsion in integer homology that grows with .

As far as we know at the moment, no one has written down any improvements to the trivial bounds on , that . Any progress on this problem will require new tools to handle torsion in random homology, and will no doubt be of interest in both geometric group theory and stochastic topology.

Nati Linial and Roy Meshulam defined a certain kind of random two-dimensional simplicial complex, and found the threshold for vanishing of homology. Their theorem is in some sense a perfect homological analogue of the classical Erdős–Rényi characterization of the threshold for connectivity of the random graph.

Linial and Meshulam’s definition was as follows. is a complete graph on vertices, with each of the triangular faces inserted independently with probability , which may depend on . We say that almost always surely (a.a.s) has property if the probability that tends to one as .

Nati Linial and Roy Meshulam showed that if is any function that tends to infinity with and if then a.a.s , and if then a.a.s .

(This result was later extended to arbitrary finite field coefficients and arbitrary dimension by Meshulam and Wallach. It may also be worth noting for the topologically inclined reader that their argument is actually a cohomological one, but in this setting universal coefficients gives us that homology and cohomology are isomorphic vector spaces.)

Eric Babson, Chris Hoffman, and I found the threshold for vanishing of the fundamental group to be quite different. In particular, we showed that if is any constant and then a.a.s. and if then a.a.s. . The harder direction is to show that on the left side of the threshold that the fundamental group is nontrivial, and this uses Gromov’s ideas of negative curvature. In particular to show that the is nontrivial we have to show first that it is a hyperbolic group.

[I want to advertise one of my favorite open problems in this area: as far as I know, nothing is known about the threshold for , other than what is implied by the above results.]

I was thinking recently about a cubical analogue of the Linial-Meshulam set up. Define to be the one-skeleton of the -dimensional cube with each square two-dimensional face inserted independently with probability . This should be the cubical analogue of the Linial-Mesulam model? So what are the thresholds for the vanishing of and ?

I just did some “back of the envelope” calculations which surprised me. It looks like must be much larger (in particular bounded away from zero) before either homology or homotopy is killed. Here is what I think probably happens. For the sake of simplicity assume here that is constant, although in realty there are terms that I am suppressing.

(1) If then a.a.s , and if then a.a.s .

(2) If then a.a.s. , and if then a.a.s. .

Perhaps in a future post I can explain where the numbers and come from. Or in the meantime, I would be grateful for any corroborating computations or counterexamples.

If a two-dimensional simplicial complex has vertices and faces, does it necessarily contain an embedded torus?

I want to advertise this question to a wider audience, so I’ll explain first why I think it is interesting.

First of all this question makes sense in the context of Turán theory, a branch of extremal combinatorics. The classical Turán theorem gives that if a graph on vertices has more than edges then it necessarily contains a complete subgraph on vertices. This is tight for every and .

One could ask instead how many edges one must have before there is forced to be a cycle subgraph, where it doesn’t matter what the length of the cycle is. This is actually an easier question, and it is easy to see out that if one has edges there must be a cycle.
It also seems more natural, in that it can be phrased topologically: how many edges must be added to vertices before we are forced to contain an embedded image of the circle?

What is the right two-dimensional analogue of this statement? In particular, is there a constant such that a two-dimensional simplicial complex with vertices and at least two-dimensional faces must contain an embedded sphere ? If so, then this is essentially best possible. By taking a cone over a complete graph on vertices, one constructs a two-complex on vertices with faces and no embedded spheres. Without having thought about it at all, I am not sure how to do better.

In any case, the corresponding question for torus seems more interesting, but for different reasons. In a paper with Eric Babson and Chris Hoffman we looked at the fundamental group of random two-complexes, as defined by Linial and Meshulam, and found the rough threshold for vanishing of the fundamental group. To show that the fundamental group was nontrivial when the number of faces was small required a lot of work — in particular, in order to apply Gromov’s local-to-global method for hyperbolicity, we needed to prove that the space was locally negatively curved, and this meant classifying the homotopy type of subcomplexes up to a large but constant size.

It turned out that the small subcomplexes were all homotopy equivalent to wedges of circles, spheres, and projective planes. In particular, we show that there are not any torus subcomplexes, at least not of bounded size. (Linial may have recently shown that there are not embedded tori, even of size tending to infinity with .) On the other hand, just on the other side of the threshold embedded tori abound in great quantity. It is interesting that something similar happens in the density random groups of Gromov — that the threshold for vanishing of the density random group corresponds to the presence of tori subcomplex in the naturally associated two-complex. It is not clear to me if this is a general phenomenon, coming geometrically from the fact that a torus admits a flat metric.

Some of the great successes of the probabilistic method in combinatorics have been in existence proofs when constructions are hard or impossible to come by. It would be nice to have interesting or extremal topological examples produced this way. Nati’s question suggests an interesting family of extremal problems in topological combinatorics, and it might make sense that in certain cases, random simplicial complexes have nearly maximally many faces for avoiding a particular embedded subspace.

Update: Nati pointed me to the paper Sós, V. T.; Erdos, P.; Brown, W. G., On the existence of triangulated spheres in $3$-graphs, and related problems. Period. Math. Hungar. 3 (1973), no. 3-4, 221–228.
Here it is shown that is the right answer for the sphere. Their lower bound is constructive, based on projective planes over finite fields. Nati said that being initially unaware of this paper, he found a probabilistic proof that works just as well as a lower bound for every fixed 2-manifold. So it seems that the main problem here is to find a matching upper bound for the torus.

A simplicial complex is said to be flag if it is the clique complex of its underlying graph. In other words, one starts with the graph and add all simplices of all dimensions that are compatible with this -skeleton. A subcomplex of a flag complex is said to be induced if it is flag, and if whenever vertices and is an edge of , we also have that is an edge of .

Does there exist a flag simplicial complex with countably many vertices, such that the following extension property holds?

[Extension property] For every finite or countably infinite flag simplicial complex and vertex , and for every embedding of as an induced subcomplex , can be extended to an embedding of as an induced subcomplex.

It turns out that such a does exist, and it is unique up to isomorphism (both combinatorially and topologically). Other interesting properties of immediately follow.

– The automorphism group of acts transitively on -dimensional faces for every .

– Deleting any finite number of vertices or edges of and the accompanying faces does not change its homeomorphism type.

Here is an easy way to describe . Take countably many vertices, say labeled by the positive integers. Choose a probability such that , and for each pair of integers , connect to by an edge with probability . Do this independently for every edge.

This is sometimes called the Rado graph, and because it is unique up to isomorphism (and in particular because it does not depend on ) it is sometimes also called the random graph. It is also possible to construct the Rado graph purely combinatorially, without resorting to probability. The I have in mind is of course just the clique complex of the Rado graph.

We can filter the complex by setting to be the induced subcomplex on all vertices with labels , and this allows us to ask more refined questions. (Now the choice of affects the asymptotics, so we assume .) From the perspective of homotopy theory, is not a particularly interesting complex; it is contractible. (This is an exercise, one should check this if it is not obvious!) However, has interesting topology.

As , the probability that is contractible is going to . It was recently shown that has asymptotically almost surely (a.a.s.) at least nontrivial homology groups, concentrated around dimension . For comparison, the dimension of is .

I think one can probably show using the techniques from this paper that there is a.a.s. no nontrivial homology above dimension or below dimension . It is still not clear (at least to me) what happens between dimensions and . It seems that a naive Morse theory argument can give that the expected dimension of homology is small in this range, but to show that it is zero would take a more refined Morse function. Perhaps a good topic for another post would be “Morse theory in probability.”

Another question: given a non-contractible induced subcomplex (say an embedded -dimensional sphere) on a set of vertices, how many vertices should one expect to add to before becomes contractible in the larger induced subcomplex? For example, it seems that once you have added about vertices, it is reasonably likely that one of these vertices induces a cone over , but is it possible that the subcomplex becomes contractible with far fewer vertices added?

I recently posted an article to the arXiv, on a problem suggested by Persi Diaconis. “Hard discs in a box” is a classical setting in statistical physics. One very simple model of matter is as non-overlapping (or hard) discs of radius , bouncing around inside a unit square. This is a very well studied model in computational statistical physics, although not much seems to be known so far mathematically. In particular, the statistical physicists observe experimentally that there is a phase transition of some kind when .

The details of what exactly it means that there is a phase transition will have to be left for another post. But what is relevant for us here is how they sample a uniformly random configuration of hard discs, via the Metropolis algorithm. (Please see the recent survey article by Persi Diaconis in the Bulletin of the AMS.) Basically, one starts with the discs in a reasonable starting configuration, say evenly spaced along some lattice, and then one applies Metropolis, which means the following. Choose one of the discs uniformly randomly, and make a small perturbation of its position. If this results in overlapping discs, or going out of the square, reject the move, otherwise accept it.

Under certain hypotheses on your space, one can show that this Markov chain converges to the uniform distribution on the space of all possible configurations. But to know this, one must know that the Markov chain is irreducible, and in particular that the configuration space is path connected.

Diaconis, Lebeau, and Michel observed that there is some constant such that if , the discs can always be freely rearranged, and in particular that the Metropolis Markov chain is irreducible. The real point of their article was to put useful upper bounds on the mixing time of Metropolis in general, and then they applied it to the hard discs setting as a motivating example. It would be nice if the same holds even in the range of the phase transition; in other words we would like to replace in the above statement with .

However, their result is essentially best possible. In my recent preprint, I gave examples of stable configurations (meaning that no single disc can move) with . Here is a picture that should give some feeling for the main idea behind the construction.

There is still quite a lot to do here. As Persi points out in his survey article, these configuration spaces of hard discs are naturally endowed with a topological structure, as well as probability measure. In his survey article, he says, “very, very little is known about the topology of these spaces.”

At a workshop on, “Topological complexity of random sets,” this past week at the American Institute of Mathematics, Yuliy Baryshnikov, Peter Bubenik, and I were able to establish a connection to classical algebraic topology. In particular, if , then the configuration space is not only path connected, but it is homotopy equivalent to the classical configuration space of labeled points in the plane. It is fairly clear that this happens for some sufficiently small radius, but as far as we know, no one had put any quantitative bounds on it yet.

In another article, in preparation, I discuss the combinatorial aspects of these configuration spaces, and give an example of what I call a “transitionary” configuration, as a first attempt to give a heuristic explanation of the phase transition near . The disc packing example illustrated above might be more interesting to discrete geometers than to probabilists, since one would like to believe that examples like this are rather rare, in a measure theoretic sense. Nevertheless, we believe that deeply understanding the various geometric and topological features of these spaces, and how they change as varies, might be a first step in explaining the phase transition mathematically.